Spring 2015

Friday, May 22, 2015

Title:Automorphism bases for the lattice of computably enumerable
vector spaces

Abstract: The lattice L of computably enumerable vector spaces
and its factor lattice modulo finite dimension, L*, have been extensively studied. Both lattices have complemented
elements, also called decidable spaces. Ash and Downey showed that every computably
enumerable vector space is a direct sum of two decidable spaces. Hence every automorphism of L
is completely determined by its action on the decidable spaces. We will talk
about other classes of computably enumerable vector spaces with this property.

Friday, April 3, 2015

Abstract:
I was asked to
give a “speculative” math talk. So, I'll discuss the question of just how
much “normal, finitary” math the Gödel incompleteness
phenomenon might infest. I’ll first survey the types of independence and undecidability results that are known, and explain why in
my view, none of them give a fully satisfactory
answer. I’ll then speculate about the question, which I’m often asked, of
whether the P vs. NP problem might turn out to be formally undecidable.
Finally, I’ll discuss the Busy Beaver function, and its amazing ability to
“concretize” questions of mathematical logic. I’ll mention some ongoing work
with Adam Yedidia that aims to construct a small
Turing machine whose (non-)halting is provably
independent of the ZFC axioms.

Friday, February 20, 2015

2:30–3:30p.m.

Speaker:Hakim Walker GWU

Place:Government Hall, Room 101

Title:Computable categoricity of a class of graphs

Abstract: We will discuss some
computability-theoretic properties of graphs. Two computable graphs are said to
be computably isomorphic if there is a computable isomorphism between them, and
a computable graph is called computably categorical if every two computable
presentations of the graph are computably isomorphic. We will discuss a
characterization of the strongly locally finite graphs that are computably
categorical, and present some examples.

Friday, February 13, 2015

2:30–3:30p.m.

Speaker:Trang Ha, GWU

Place:Government Hall, Room 101

Title:Orderable but not
computably orderable groups

Abstract: An order (also called a
bi-order) on a group is a total ordering of the elements of its domain, which
is both left-invariant and right-invariant with
respect to the group operation. We will present a construction of a computable
orderable group, which does not have a computable order.

Friday, February 6, 2015

2:30–3:30p.m.

Speaker:Leah Marshall, GWU

Place:Government Hall, Room 101

Title:Complexity of relations
and the arithmetical hierarchy

Abstract:
One way of
examining computability-theoretic complexity of various natural properties
(relations) of computable structures is to examine the complexity of the formulas
that define the properties. In this talk, we will discuss some examples of the
arithmetical complexity of various objects, pulling from the more standard and
classic examples, as well as describing some examples from current research. We
will review arithmetical hierarchy. The talk will be accessible to all
graduate students.

Friday, January 23, 2015

Title:Strictness, laziness, and effectsAbstract: Computer scientists want to
be able to write down programs and have them correspond to a single
computation. Sadly, most programming languages don't allow this: a single
program might be read multiple ways. The standard strategies for fixing this,
strictness and laziness, have been used since the 1930s, but continue to be
poorly understood. The questions of why most languages have multiple readings,
and why strictness and laziness are the natural solutions, remain mostly
unanswered. We still lack even a formal definition of laziness that is not
language-dependent.

This talk focuses on recent
work that attempts to provide a formal definition and answer these questions.
We achieve this by looking through the lens of effects, which allow us to
classify computations in input-independent ways.

Colloquium

Friday, January 16, 2015

Title:The rich structure of modular lattices arising from
computably enumerable vector spacesAbstract: Post’s problem
in computability theory, dating back to 1944, is whether there exist a computably enumerable (c.e.)
Turing degree that is neither computable nor the degree of the
halting problem. His strategy was to find a non-computable co-infinite c.e. set with a “thin” complement.
The original notion of thinness was called“simplicity.”
A c.e. set the complement of
which is infinite but has no infinite c.e. subset is called simple. Different, ever-stronger, notions
of thinness were defined, but none of them gave a solution to Post’s problem.
These notions, however, revealed some fascinating structural properties of the
lattice E of c.e. sets under
inclusion, and its factor lattice E* (E modulo finite sets). In the talk we
will introduce the lattice L of c.e. subspaces of the fully algorithmic infinite dimensional
vector space and its factor lattice L* (L modulo finite dimension). We will
explore some important similarities and differences between E* and L*.

Thursday, January 15, 2015

Abstract: In this talk we will start with some standard ways of
obtaining nonstandard structures. We will then introduce the notion of cohesive
power B of a computable structure A (see [1]) over a cohesive set R. We will prove certain connections
between satisfaction of different formulas and sentences in the original model A and its cohesive power B.

Friday, September 26, 2014

Abstract:Many
classes of computable structures can be enumerated computably. For example, one
readily gives a uniformly computable list of all computable linear orders,
simply by enumerating the c.e. subsets
of a single computable dense linear order. Of course, this list includes
infinitely many computable copies of each computable linear order. To give a
computable classification (up to isomorphism) of these linear orders would
require computing such a list so that no two linear orders on the list are
(classically) isomorphic to each other. This is known to be impossible.

The paradigm of a computable
classification was given by Friedberg, who produced a uniformly computable
listing of all c.e. sets,
with no set appearing more than once in the listing. That is, he gave a
computable classification of the c.e. sets up to equality. We apply his method to yield a
computable classification of the computable algebraic fields, up to (classical)
isomorphism. We also follow Goncharov and Knight in
showing that certain other classes have no computable classification.

Finally, we give a 0'-computable classification of the computable equivalence
structures. This result, which extends (and uses) more work of Goncharov and Knight, means that there is a uniformly 0'-computable listing of all computably
presentable equivalence structures, with no isomorphisms
between any two distinct structures on the list; however, the structures on the
list are only 0'-computable, not
necessarily computable. We conjecture that there is no computable
classification of the computable equivalence structures.

Friday, September 19, 2014

Abstract:A partial computable
injection structure is a mathematical structure consisting of a computable set
of natural numbers and a partial computable, injective (1-1) function.
The “shape” of these structures is determined by the orbits
of the elements; that is, what happens when we apply our function to an
element repeatedly. These structures can therefore be completely
classified up to isomorphism by the numbers, types, and sizes of their orbits.
However, we know that isomorphisms alone do not
necessarily preserve the computability-theoretic properties of mathematical
structures. We examine partial computable injection structures with
computable isomorphisms, and we explain what goes
wrong in structures without such computable isomorphisms.
Additionally, we do the same for partial computable injection structures under
Δ2-isomorphisms and Δ3-isomorphisms.

Logic-Topology Seminar

Friday, September 12, 2014

Abstract:For the 30th anniversary of
the Homflypt polynomial of links, I propose a new
polynomial invariant of rooted trees. I will relate this to the Kauffman
bracket (version of the Jones polynomial) and to (pre)simplicial categories.